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A General Result for the Magnetoelastic Response Of Journal of the Mechanics and Physics of Solids 107 (2017) 343–364 Contents lists available at ScienceDirect Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps A general result for the magnetoelastic response of isotropic suspensions of iron and ferrofluid particles in rubber, with applications to spherical and cylindrical specimens ∗ Victor Lefèvre a, Kostas Danas b, Oscar Lopez-Pamies a, a Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, United States b LMS, C.N.R.S., École Polytechnique, Université Paris-Saclay, 91128 Palaiseau, France a r t i c l e i n f o a b s t r a c t Article history: This paper puts forth an approximate solution for the effective free-energy function de- Received 10 April 2017 scribing the homogenized (or macroscopic) magnetoelastic response of magnetorheological Revised 10 May 2017 elastomers comprised of non-Gaussian rubbers filled with isotropic suspensions of either Accepted 25 June 2017 iron or ferrofluid particles. The solution is general in that it applies to N = 2 and 3 space Available online 13 July 2017 dimensions and any arbitrary (non-percolative) isotropic suspension of filler particles. By Keywords: construction, it is exact in the limit of small deformations and moderate magnetic fields. Magnetorheological elastomers For finite deformations and finite magnetic fields, its accuracy is demonstrated by means Ferrofluid inclusions of direct comparisons with full-field simulations for two prominent cases: ( i ) isotropic sus- Magnetostriction pensions of circular particles and ( ii ) isotropic suspensions of spherical particles. Finite magnetoelastostatics With the combined objectives of demonstrating the possible benefits of using fer- rofluid particles in lieu of the more conventional iron particles as fillers and gaining insight into recent experimental results, the proposed homogenization-based constitutive model is deployed to generate numerical solutions for boundary-value problems of both fundamental and practical significance: those consisting of magnetorheological elastomer specimens of spherical and cylindrical shape that are immersed in air and subjected to a remotely applied uniform magnetic field. It is found that magnetorheological elastomers filled with ferrofluid particles can exhibit magnetostrictive capabilities far superior to those of magnetorheological elastomers filled with iron particles. The results also reveal that the deformation and magnetic fields are highly heterogenous within the specimens and strongly dependent on the shape of these, specially for magnetorheological elas- tomers filled with iron particles. From an applications perspective, this evidence makes it plain that attempts at designing magnetrostrictive devices based on magnetorheological elastomers need to be approached, in general, as structural problems, and not simply as materials design problems. ©2017 Elsevier Ltd. All rights reserved. 1. Introduction Ostensibly due to the renewed experimental impetus started during the 1990s (see, e.g., Ginder et al., 1999; Jolly et al., 1996; Shiga et al., 1995 ), increasing efforts have been devoted by the mechanics community to construct continuum mod- ∗ Corresponding author. E-mail addresses: [email protected] (V. Lefèvre), [email protected] (K. Danas), [email protected] (O. Lopez-Pamies). http://dx.doi.org/10.1016/j.jmps.2017.06.017 0022-5096/© 2017 Elsevier Ltd. All rights reserved. 344 V. Lefèvre et al. / Journal of the Mechanics and Physics of Solids 107 (2017) 343–364 Fig. 1. Schematic of a magnetorheological elastomer in its ground configuration depicting its underlying microstructure comprised of a random isotropic suspension of either (a) iron particles or (b) ferrofluid particles firmly embedded in a rubber matrix. The magnetoelastic behaviors of the rubber matrix and the particles are characterized by free-energy functions W m and W p . The macroscopic magnetoelastic behavior of the magnetorheological elastomer is characterized by the effective free-energy function W . els capable of describing the magnetoelastic response of magnetorheological elastomers under finite deformations (involv- ing arbitrary finite strains and rigid rotations) and finite magnetic fields. These efforts can be roughly classified into two categories: ( i ) top-down or phenomenological approaches in which macroscopic free energies are postulated based on macroscopic experimental observations (see, e.g., Bustamante et al., 2011; Danas et al., 2012; Dorfmann and Ogden, 2005; Kankanala and Triantafyllidis, 2004; Pelteret et al., 2016; Saxena et al., 2015 ) and ( ii ) bottom-up or homogenization ap- proaches in which macroscopic free energies are derived based on the underlying microscopic behavior (see, e.g., Borcea and Bruno, 2001; Corcolle et al., 2008; Galipeau and Ponte Castaeda, 2012; Galipeau and Ponte Castañeda, 2013; Liu et al., 2006; Zhou and Shin, 2005 ). While the practical challenges of carrying out experiments that test the material (and not the structural) response of specimens over wide ranges of finite deformations and finite magnetic fields have curtailed the ad- vancement of phenomenological models, the intrinsic mathematical challenges of carrying out the homogenization limit of the equations of magnetoelastostatics have hindered the construction of homogenization-based models. In this context, leveraging a recent result of Lefèvre and Lopez-Pamies (2017a, b) within the mathematically analogous setting of electroelastostatics, the fundamental object of this work is to put forth a homogenization-based macroscopic free energy that describes the finite magnetoelastic response of isotropic magnetorheological elastomers under arbitrary magne- tomechanical loadings. The focus is on isotropic magnetorheological elastomers —in both N = 2 and 3 space dimensions — comprised of a non-Gaussian rubber matrix isotropically filled with either iron or ferrofluid particles; see Fig. 1 for a schematic. By deploying the constructed free energies, an additional practical objective of this work is also to provide in- sight into the merits of using ferrofluid particles in lieu of the more conventional iron particles as fillers. A further practical objective is to scrutinize experiments available in the literature on magnetorheological elastomers containing iron particles. This is accomplished by carrying out finite-element simulations of representative experiments making use of the constructed free energies to model the magnetoelastic behavior of the specimens. To put the present work in perspective, we remark that existing analytical homogenization (exact or approximate) results for isotropic magnetorheological elastomers are restricted to the asymptotic context of small deformations, save for an ap- proximate result due to Galipeau and Ponte Castaeda (2012) in N = 2 space dimensions that is valid for finite deformations. These authors made use of a partial decoupling approximation ( Ponte Castañeda and Galipeau, 2011 ) together with an earlier result of Lopez-Pamies and Ponte Castañeda (2006) to construct an estimate for the macroscopic free energy of an isotropic incompressible elastic matrix reinforced by an isotropic suspension of circular magnetizable particles that are mechanically rigid. It is also fitting to remark that computational homogenization results have been recently reported in the literature for rubber filled with periodic square/hexagonal arrays ( Galipeau et al., 2014; Javili et al., 2013; Keip and Rambausek, 2016 ) and approximately isotropic distributions ( Danas, 2017; Kalina et al., 2016 ) of circular particles in N = 2 space dimensions and with periodic cubic arrays of spherical particles ( Javili et al., 2013; Miehe et al., 2016 ) in N = 3 space dimensions. These computational results pertain to rubber matrices featuring high compressibility (presumably in order to avoid numerical complications such as volumetric locking). We also remark that neither theoretical nor experimental studies on magne- torheological elastomers containing ferrofluid filler particles appear to have been reported in the literature; see, however, the recent works of Lopez-Pamies (2014) , Lefèvre and Lopez-Pamies (2017b) , Barlett et al. (2017) , and references therein for intimately related studies of dielectric elastomers filled with liquid-metal inclusions. The presentation of the work is organized as follows. We begin in Section 2 by formulating the problem in N = 2 and 3 space dimensions that defines the macroscopic magnetoelastic response of isotropic incompressible non-Gaussian rubber, filled with an arbitrary isotropic suspension of deformable magnetizable particles whose magnetization may possibly saturate, under finite deformations and finite electric fields. In Section 3 , we present an approximate solution for the problem formulated in Section 2 for two types of filler particles: iron (modelled as mechanically rigid) and ferrofluids (modeled as mechanically liquid-like, that is, incompressible and of vanishingly small shear stiffness). For the case of N = 3 space dimensions, the solution corresponds to a recasting, mutatis mutandis , of the solution recently derived by Lefèvre and Lopez-Pamies (2017b ) within the mathematically analogous setting of electroelastostatics. The solution for N = 2 space dimensions corresponds to a generalization of such a solution beyond N = 3 . We devote Sections 4 and 5 to spelling out the specializations of the general solution presented in Section 3 to the basic cases of isotropic
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